A000588 -id:A000588 - cp's OEIS frontend

A026029 Number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n) = 3. Also T(2n,n), where T is defined in A026022.

1, 2, 6, 20, 69, 242, 858, 3068, 11050, 40052, 145996, 534888, 1968685, 7276050, 26993490, 100490220, 375287550, 1405622460, 5278838100, 19873977240, 74994427170, 283595947284, 1074568266756, 4079184055640, 15511924233204, 59083160374952, 225384613313944

Offset: 0

Clark Kimberling

nonn

Hankel transform is A008619(n+1). - Paul Barry, May 11 2009

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.

CoefficientList[Series[(1 - 2*x)*(-1 + Sqrt[1 - 4*x] + 2*x)^2 / (4*x^4), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 03 2019 *)

Expansion of (1+x^2*C^4)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
a(n) = Sum_{k=0..n} C(n, k)*Sum_{i=0..k} C(k, 2i)*A000108(i+1). - Paul Barry, Jul 18 2003
a(n) = Sum_{k=0..3} A039599(n,k) = A000108(n) + A000245(n) + A000344(n) + A000588(n) = A026012(n) + A000588(n). - Philippe Deléham, Nov 12 2008
a(n) = C(2n,n) - C(2n,n-4). - Paul Barry, May 11 2009
Conjecture: (n+4)*a(n) + 6*(-n-2)*a(n-1) + 4*(2*n-1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
a(n) ~ 4^(n+2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 03 2019
E.g.f.: exp(2*x)*(BesselI(0, 2*x) - BesselI(4, 2*x)). - Stefano Spezia, Jan 17 2024

A050155 Triangle T(n,k), k>=0 and n>=1, read by rows defined by: T(n,k) = (2k+3)*binomial(2n,n-k-1)/(n+k+2).

Original entry on oeis.org

1, 3, 1, 9, 5, 1, 28, 20, 7, 1, 90, 75, 35, 9, 1, 297, 275, 154, 54, 11, 1, 1001, 1001, 637, 273, 77, 13, 1, 3432, 3640, 2548, 1260, 440, 104, 15, 1, 11934, 13260, 9996, 5508, 2244, 663, 135, 17, 1, 41990, 48450, 38760, 23256, 10659, 3705, 950, 170, 19, 1

Offset: 1

Clark Kimberling

T(n-2k-1,k) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 2k+2 (cf. Zoran Sunic reference) . - Benoit Cloitre, Oct 07 2003
Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=k+1 . - Herbert Kociemba, May 24 2004
Number of standard tableaux of shape (n+k+1, n-k-1). - Emeric Deutsch, May 30 2004
Riordan array (c(x)^3,xc(x)^2) where c(x) is the g.f. of A000108. Inverse array is A109954. - Paul Barry, Jul 06 2005

			    1;
    3,   1;
    9,   5,   1;
   28,  20,   7,  1;
   90,  75,  35,  9,  1;
  297, 275, 154, 54, 11, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
R. K. Guy, Catwalks, Sansteps and Pascal Pyramids, J. Integer Seq., Vol. 3 (2000), #00.1.6
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
A. Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math 14 (1957), 405-414.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
_Zoran Sunic_, Self-Describing Sequences and the Catalan Family Tree, Electronic Journal of Combinatorics, 10 (2003) #N5.

First columns: A000245, A000344, A000588, A001392, A000589, A000590.

Cf. A000108, A001791 (row sums), A050144.

T:= (n, k)->  (2*k+3)*binomial(2*n, n-k-1)/(n+k+2):
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Jan 19 2013

T[n_, k_] :=  (2*k + 3)*Binomial[2*n, n - k - 1]/(n + k + 2);
Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, May 21 2016 *)

Sum_{ k = 0, .., n-1} T(n, k) = binomial(2n, n-1) = A001791(n).
G.f. of column k: x^(k+1)*C^(2*k+3) where C = (1-(1-4*x)^(1/2))/(2*x) is the g.f. of Catalan numbers A000108. - Philippe Deléham, Feb 03 2004
T(n, k) = A039599(n, k+1) = A009766(n+k+1, n-k-1) = A033184(n+k+2, 2k+3) . - Philippe Deléham, May 28 2005
Sum_{k>= 0} T(m, k)*T(n, k) = A000108(m+n) - A000108(m)*A000108(n). - Philippe Deléham, May 28 2005
T(n, k)=(2k+3)binomial(2n+2, n+k+2)/(n+k+3)=C(2n+2, n+k+2)-C(2n+2, n+k+3) [offset (0, 0)]. - Paul Barry, Jul 06 2005

Edited by Philippe Deléham, May 22 2005

A026014 a(n) = number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n) = 6. Also a(n) = T(2n,n-2), where T is the array defined in A026009.

Original entry on oeis.org

1, 6, 28, 119, 483, 1911, 7448, 28764, 110466, 422807, 1615152, 6163885, 23514855, 89714835, 342411120, 1307613480, 4997082510, 19111589280, 73154916744, 280265589198, 1074685552094, 4124573481446, 15843809385168, 60914041121640

Offset: 2

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 2..1000

First differences of A000588.

Cf. A000108, A026009.

[Binomial(2*n, n-2) - Binomial(2*n, n-5): n in [2..30]]; // G. C. Greubel, Mar 19 2021

Table[Binomial[2*n, n-2] - Binomial[2*n, n-5], {n, 2, 30}] (* G. C. Greubel, Mar 19 2021 *)

[binomial(2*n, n-2) - binomial(2*n, n-5) for n in (2..30)] # G. C. Greubel, Mar 19 2021

-(n-2)*(n+5)*(n+23)*a(n) +(-n^3+127*n^2+188*n-432)*a(n-1) +2*(n-1)*(2*n-3)*(5*n-24)*a(n-2) = 0. - R. J. Mathar, Jun 20 2013
From G. C. Greubel, Mar 19 2021: (Start)
G.f.: (1-x)*(1 -7*x +14*x^2 -7*x^3 -(1 -5*x +6*x^2 -x^3)*sqrt(1-4*x))/(2*x^5).
G.f.: (1-x)*x^2*C(x)^7, where C(x) is the g.f. of the Catalan numbers (A000108).
E.g.f.: exp(2*x)*(BesselI(2, 2*x) - BesselI(5, 2*x)).
a(n) = binomial(2*n, n-2) - binomial(2*n, n-5) = A026009(2*n, n-2).
a(n) = 1 if n = 2 else f(n) - f(n-1), where f(n) = Sum_{j=0..n-2} C(n-j-2)*(C(j+5) -4*C(j+4) +3*C(j+3)) and C(n) are the Catalan numbers. (End)
From G. C. Greubel, Mar 22 2021: (Start)
a(n) = C(n+4) -6*C(n+3) +11*C(n+2) -7*C(n+1) +C(n).
a(n) = 21*(n*(n-1)*(n^2+n+4)/((n+2)*(n+3)*(n+4)*(n+5)))*C(n), where C(n) are the Catalan numbers. (End)

A355173 The Fuss-Catalan triangle of order 1, read by rows. Related to binary trees.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 4, 9, 14, 0, 1, 5, 14, 28, 42, 0, 1, 6, 20, 48, 90, 132, 0, 1, 7, 27, 75, 165, 297, 429, 0, 1, 8, 35, 110, 275, 572, 1001, 1430, 0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 0, 1, 10, 54, 208, 637, 1638, 3640, 7072, 11934, 16796

Offset: 0

Peter Luschny, Jun 25 2022

The Fuss-Catalan triangle of order m is a regular, (0, 0)-based table recursively defined as follows: Set row(0) = [1] and row(1) = [0, 1]. Now assume row(n-1) already constructed and duplicate the last element of row(n-1). Next apply the cumulative sum m times to this list to get row(n). Here m = 1. (See the Python program for a reference implementation.)

This definition also includes the classical Fuss-Catalan numbers, since T(n, n) = A000108(n), or row 2 in A355262. For m = 2 see A355172 and for m = 3 A355174. More generally, for n >= 1 all Fuss-Catalan sequences (A355262(n, k), k >= 0) are the main diagonals of the Fuss-Catalan triangles of order n - 1.

			Table T(n, k) begins:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1, 2]
  [3] [0, 1, 3,  5]
  [4] [0, 1, 4,  9,  14]
  [5] [0, 1, 5, 14,  28,  42]
  [6] [0, 1, 6, 20,  48,  90,  132]
  [7] [0, 1, 7, 27,  75, 165,  297, 429]
  [8] [0, 1, 8, 35, 110, 275,  572, 1001, 1430]
  [9] [0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862]
Seen as an array reading the diagonals starting from the main diagonal:
  [0] 1, 1, 2,  5,  14,   42,  132,   429,  1430,   4862,   16796, ...  A000108
  [1] 0, 1, 3,  9,  28,   90,  297,  1001,  3432,  11934,   41990, ...  A000245
  [2] 0, 1, 4, 14,  48,  165,  572,  2002,  7072,  25194,   90440, ...  A099376
  [3] 0, 1, 5, 20,  75,  275, 1001,  3640, 13260,  48450,  177650, ...  A115144
  [4] 0, 1, 6, 27, 110,  429, 1638,  6188, 23256,  87210,  326876, ...  A115145
  [5] 0, 1, 7, 35, 154,  637, 2548,  9996, 38760, 149226,  572033, ...  A000588
  [6] 0, 1, 8, 44, 208,  910, 3808, 15504, 62016, 245157,  961400, ...  A115147
  [7] 0, 1, 9, 54, 273, 1260, 5508, 23256, 95931, 389367, 1562275, ...  A115148

A000108 (main diagonal), A000245 (subdiagonal), A002057 (diagonal 2), A000344 (diagonal 3), A000027 (column 2), A000096 (column 3), A071724 (row sums), A000958 (alternating row sums), A262394 (main diagonal of array).
Variants: A009766 (main variant), A030237, A130020.
Cf. A123110 (triangle of order 0), A355172 (triangle of order 2), A355174 (triangle of order 3), A355262 (Fuss-Catalan array).
Cf. A115144, A115145, A000588, A115147, A115148.

from functools import cache
from itertools import accumulate
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 1]
    row = Trow(n - 1) + [Trow(n - 1)[n - 1]]
    return list(accumulate(row))
for n in range(11): print(Trow(n))

The general formula for the Fuss-Catalan triangles is, for m >= 0 and 0 <= k <= n:
FCT(n, k, m) = (m*(n - k) + m + 1)*(m*n + k - 1)!/((m*n + 1)!*(k - 1)!) for k > 0 and FCT(n, 0, m) = 0^n. The case considered here is T(n, k) = FCT(n, k, 1).
T(n, k) = (n - k + 2)*(n + k - 1)!/((n + 1)!*(k - 1)!) for k > 0; T(n, 0) = 0^n.
The g.f. of row n of the FC-triangle of order m is 0^n + (x - (m + 1)*x^2) / (1 - x)^(m*n + 2), thus:
T(n, k) = [x^k] (0^n + (x - 2*x^2)/(1 - x)^(n + 2)).

A050185 T(2n+7,n), array T as in A051168; a count of Lyndon words.

Original entry on oeis.org

0, 1, 5, 22, 91, 364, 1428, 5537, 21318, 81719, 312455, 1193010, 4552275, 17368680, 66284554, 253086480, 966955410, 3697182450, 14147884842, 54185826156, 207712333598, 796937116661, 3060338457400, 11762344331920

Offset: 0

Clark Kimberling

nonn

Is a(n) = A000588(n+3)/7 if n is not a multiple of 7? R. J. Mathar, Jul 24 2012

Index entries for sequences related to Lyndon words

A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 9, 4, 1, 14, 28, 14, 6, 1, 42, 90, 48, 27, 7, 1, 132, 297, 165, 110, 35, 9, 1, 429, 1001, 572, 429, 154, 54, 10, 1, 1430, 3432, 2002, 1638, 637, 273, 65, 12, 1, 4862, 11934, 7072, 6188, 2548, 1260, 350, 90, 13, 1, 16796, 41990, 25194, 23256, 9996, 5508, 1700, 544, 104, 15, 1

Offset: 0

Philippe Deléham, Feb 01 2014

Row sums are A109262(n+1).

			Triangle begins:
    1;
    1,   1;
    2,   3,   1;
    5,   9,   4,   1;
   14,  28,  14,   6,  1;
   42,  90,  48,  27,  7, 1;
  132, 297, 165, 110, 35, 9, 1;
Production matrix is:
  1...1
  1...2...1
  0...1...1...1
  0...1...1...2...1
  0...0...0...1...1...1
  0...0...0...1...1...2...1
  0...0...0...0...0...1...1...1
  0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...1...1...1
  0...0...0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...0...0...1...1...1
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns: A000108 (k=0), A000245 (k=1), A002057 (k=2), A003517 (k=3), A000588 (k=4), A001392 (k=5), A003519 (k=6), A090749 (k=7), A000590 (k=8).
Cf. A026674, A026726, A032766, A109262.
Cf. A000045, A039599, A106566.

F:=Factorial;
A236843:= func< n,k | (1/4)*(6*k+5-(-1)^k)*F(2*n-Floor(k/2))/(F(n-k)*F(n+Floor((k+1)/2)+1)) >;
[A236843(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 13 2022

T[n_, k_]:= (1/4)*(6*k+5-(-1)^k)*(2*n-Floor[k/2])!/((n-k)!*(n+Floor[(k+1)/2]+1)!);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 13 2022 *)

T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - (k\2))!/((n-k)!*(n + (k+1)\2 + 1)!) \\ Andrew Howroyd, Jan 04 2023

F=factorial
def A236843(n,k): return (1/2)*(3*k+2+(k%2))*F(2*n-(k//2))/(F(n-k)*F(n+((k+1)//2)+1))
flatten([[A236843(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 13 2022

G.f. for the column k (with zeros omitted): C(x)^A032766(k+1) where C(x) is g.f. for Catalan numbers (A000108).
Sum_{k=0..n} T(n,k) = A109262(n+1).
Sum_{k=0..n} T(n+k,2k) = A026726(n).
Sum_{k=0..n} T(n+1+k,2k+1) = A026674(n+1).
T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - floor(k/2))!/((n-k)!*(n + floor((k+1)/2) + 1)!). - G. C. Greubel, Jun 13 2022

A050145 T(n,k)=M(2n,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 5, 4, 5, 1, 14, 14, 20, 7, 1, 42, 48, 75, 35, 9, 1, 132, 165, 275, 154, 54, 11, 1, 429, 572, 1001, 637, 273, 77, 13, 1, 1430, 2002, 3640, 2548, 1260, 440, 104, 15, 1, 4862, 7072, 13260, 9996, 5508, 2244, 663, 135, 17, 1

Offset: 0

Clark Kimberling

First 7 columns of T are A000108, A002057, A000344, A000588, A001392, A000589, A000590.

			Rows: {0}; {1,0}; {2,1,1}; ...

A158483 Triangle read by rows: T(n,k) = (4k+3)/(n+2k+2)*binomial(2n,n+2k+1).

Original entry on oeis.org

0, 1, 3, 9, 1, 28, 7, 90, 35, 1, 297, 154, 11, 1001, 637, 77, 1, 3432, 2548, 440, 15, 11934, 9996, 2244, 135, 1, 41990, 38760, 10659, 950, 19, 149226, 149226, 48279, 5775, 209, 1, 534888, 572033, 211508, 31878, 1748, 23, 1931540, 2187185, 904475, 164450

Offset: 0

Peter Bala, Mar 20 2009

This triangle forms a companion to A119245.
Combinatorial interpretations of T(n,k):
1) The number of standard tableaux of shape (n-2*k-1,n+2*k+1).
2) The entries in column k are (with an offset of 2*k+1) the number of n-th generation vertices in the tree of sequences with unit increase labeled by 4*k+2. See [Sunik, Theorem 4].

			Triangle begins
==================================
n\k|.....0.....1.....2.....3.....4
==================================
.0.|.....0
.1.|.....1
.2.|.....3
.3.|.....9.....1
.4.|....28.....7
.5.|....90....35.....1
.6.|...297...154....11
.7.|..1001...637....77.....1
.8.|..3432..2548...440....15
.9.|.11934..9996..2244...135.....1

Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).

Cf. A000245 (column 0), A000588 (column 1), A000589 (column 2), A001700 (row sums), A119245.

with(combinat): T:=(n,k) -> (4k+3)/(n+2k+2)*binomial(2n,n+2k+1): for n from 0 to 13 do seq(T(n,k),k = 0..6); end do;

T(n,k) = (4*k+3)/(n+2*k+2)*binomial(2*n,n+2*k+1).
O.g.f. y*C(y)^3/(1 - x*y^2*C(y)^4) = y + 3*y^2 + (9 + x)*y^3 + (28 + 7*x)*y^4 + ..., where C(x) = [1-(1-4*x)^(1/2)]/(2*x) is the o.g.f. for the Catalan numbers A000108.
Row sums A001700.

cp's OEIS Frontend

A026029 Number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n) = 3. Also T(2n,n), where T is defined in A026022.

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A050155 Triangle T(n,k), k>=0 and n>=1, read by rows defined by: T(n,k) = (2k+3)*binomial(2n,n-k-1)/(n+k+2).

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A026014 a(n) = number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n) = 6. Also a(n) = T(2n,n-2), where T is the array defined in A026009.

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A355173 The Fuss-Catalan triangle of order 1, read by rows. Related to binary trees.

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Python

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A050185 T(2n+7,n), array T as in A051168; a count of Lyndon words.

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A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

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A050145 T(n,k)=M(2n,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.

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A158483 Triangle read by rows: T(n,k) = (4k+3)/(n+2k+2)*binomial(2n,n+2k+1).

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Maple